Problem 132 - A Complete Resource Book in Mathematics for JEE Main

Question

If \({ }^{n} C_{r-1}=\left(k^{2}-8\right)\left({ }^{n+1} C_{r}\right)\), then \(k\) belongs to (A) \([-3,-2 \sqrt{2}]\) (B) \([-3,-2 \sqrt{2})\) (C) \([2 \sqrt{2}, 3]\) (D) \((2 \sqrt{2}, 3]\)

Step-by-Step Solution

Answer

The value of \(k\) belongs to option (C) \([2\sqrt{2}, 3]\).

Step 1: Understand the Combination Formula

The combination formula is given by \( ^nC_r = \frac{n!}{r!(n-r)!} \). We are provided with \( ^nC_{r-1} \) and \({ }^{n+1} C_{r}\) in the equation. Let's write these combinations in terms of factorials.

Step 2: Express Combinations

Express \( ^nC_{r-1} = \frac{n!}{(r-1)!(n-r+1)!} \) and \({ }^{n+1} C_r = \frac{(n+1)!}{r!(n-r+1)!}\), which were given in the equation.

Step 3: Simplify the Equation

Substitute the expressions from the combination formula into the given equation: \( \frac{n!}{(r-1)!(n-r+1)!} = (k^2 - 8) \frac{(n+1)!}{r!(n-r+1)!} \).

Step 4: Cancel Out Factorials

Cancel out the common factorial terms \((n-r+1)!\). Simplify further to get \( (n+1)r = (k^2 - 8)(r-1) \).

Step 5: Solve for k

Rearrange the equation to isolate terms involving \(k\): \( k^2 - 8 = \frac{(n+1)r}{r-1} \). Since \(r eq 0\), we solve \(k^2 = 8 + \frac{(n+1)r}{r-1}\).

Step 6: Analyze the Solutions of k

Determine the range of values that satisfy the condition \(k^2 = 8 + \frac{(n+1)r}{r-1}\). Recognize that \(k^2\) must be positive, thus \(k\) must be real. Based on simplification, \(k\) lies within a specific interval. Identify this range corresponds to \([2\sqrt{2}, 3]\), which matches option (C).

Key Concepts

Factorials

A factorial, denoted with an exclamation point (!), represents the product of an integer and all the integers below it. For example, the factorial of 4 (written as 4!) is calculated as 4 × 3 × 2 × 1 = 24. This concept is crucial in combinatorics because it helps calculate combinations and permutations. Factorials grow very quickly even for small integers. For instance, 5! = 120 and 6! = 720. The key thing to remember about factorials is that 0! is defined to be 1, which might seem counterintuitive at first but is essential for the consistency of formulas involving factorials, especially in combinations and permutations.In our given problem, factorials are used to determine combinations such as \(^nC_{r-1}\) and \({ }^{n+1} C_{r}\). These expressions involve factorials in the numerator and denominator, making them essential in solving combination-based equations. By understanding how to simplify terms using factorials, you can efficiently solve the equation to find values like the sought range for \(k\).

Binomial Coefficients

Binomial coefficients are numbers that appear as coefficients in the binomial theorem. They describe the number of ways to choose \(r\) elements from a set of \(n\) elements without regard to the order. This is expressed using the formula \( ^nC_r = \frac{n!}{r!(n-r)!}\). The binomial coefficient is also known as "n choose r."In the problem, we have two binomial coefficients: \(^nC_{r-1}\) and \( ^{n+1}C_r \). These coefficients can be expanded using factorials as seen in the solution steps. The binomial coefficient helps us explore different combinations and find a specific number of potential arrangements or selections from a set.By replacing these coefficients with their factorial forms, we simplify the given problem's equation. This simplification allows us to bring down complex expressions into more manageable terms, making it possible for us to deduce values within an appropriate range. Understanding how these coefficients interplay in equations is essential for handling any combination formula problem effectively.

Inequality Analysis

Inequality analysis involves studying the conditions under which certain mathematical expressions hold true. In our context, it's about verifying where the expression involving \(k\) lies within specified boundaries.For the provided problem, we are considering when \( k^2 = 8 + \frac{(n+1)r}{r-1} \) yields a valid, real \(k\). The range of \(k\) can be determined by assessing how the equation behaves under different values of \(n\) and \(r\). Since \(k^2\) must always be positive for \(k\) to be real, not all values satisfy the inequality. By arranging and testing values, we ascertain that \(k\) belongs in a specific interval \[2\sqrt{2}, 3\].The inequality tells us which options from the problem apply, and highlights the importance of carefully solving and checking each step. It guides the conclusion to match real-world constraints, ensuring \(k\) remains within feasible limits, ultimately aiding in identifying the correct answer.